The local-field non-embedding conjecture for completed positive Kac–Moody groups

Let $\mathcal{D}$ be a Kac–Moody root datum with indecomposable generalized Cartan matrix $A$, let $\mathbf{F}_q$ be the finite field of characteristic $p$ with $q$ elements, and let $\overline U$ be the closure of the subgroup $U$ generated by the positive root groups in any of the considered complete Kac–Moody groups. Local-field non-embedding conjecture. If the type of $A$ is neither spherical nor affine, then for every $m\geqslant 1$ and every local field $F$, there is no injective group homomorphism

$$\overline U\longrightarrow {\rm GL}_m(F).$$

This would strengthen the preceding non-linearity conjecture in the setting of representations over local fields, following the discussion of embeddings into groups of points of absolutely simple algebraic groups over non-archimedean local fields.